is true for all sufficiently large $n\gt3$. (I don't know whether or not it's actually true, but it tests as true for a bunch of small $n$ that I've looked at).

My first thought was to use the lower-bound approximation for the $n$'th prime $n[ln(n)+ln(ln(n))-1]$ as $p_n$ on the left side, and the upper-bound approximation $n[ln(n)+ln(ln(n))]$ as $p_n$ on the right side. That at least gets rid of the primes, but of course the product on the right side still isn't smooth, so I don't know where to go from there. Also, I'm not sure if the inequality would remain true with those approximations, if it's indeed true with the primes. (I did test a bunch of small values and it seemed to remain true, but again I know that means nothing.)

4 Answers
4

I happen to have a file of the first 1400 primes sitting around so I created a spreadsheet for the LHS and the RHS. The LHS is less than the RHS for twin primes 43 and higher.

Note I started the product on the RHS at $p_n=5$ since it indicates $i=3$ not $p=3$. If you start the product with $p=3$ it takes until 229 for the first violation.

Note that merely reversing the inequality will not create a viable statement as written, since the gaps between primes vary. Thus although the twin primes violate the original statement at the number given, there are many others that continue to meet the condition given.

Thank you so much. I hadn't yet tested out that far, and I'm glad to know. Starting the product at $p_n=5$ was indeed the intent--that's not a typo.
–
AnnickDec 28 '12 at 18:43

I had sort of guessed this conjecture was related to twin primes. Certainly, the LHS is least in that case...
–
Thomas AndrewsDec 28 '12 at 18:49

You did give fair warning in the statement that it might not be true. Took less time to set up the numeric test than I had already invested in the proof! Of course you should confirm for yourself in case I messed up the indexing - it was a quick and dirty test.
–
half-integer fanDec 28 '12 at 18:52

3

This doesn't really disprove the inequality, of course, it just shows that $1400$ is not "sufficiently large."
–
Thomas AndrewsDec 28 '12 at 19:14

From what I saw in my attempt to use induction (see other answer), once the statement is violated for a given prime gap, you can show that all further occurrences of that gap size will violate the inequality.
–
half-integer fanJan 2 '13 at 1:12

We can derive from the Prime Number Theorem that $p_n=n\log(n)+O(n\log(\log(n)))$. This shows that for any $\alpha\gt0$
$$
\lim_{n\to\infty}\frac{p_{\alpha n}}{p_n}=\alpha\tag{4}
$$
Furthermore, $(4)$ shows that $\lim\limits_{n\to\infty}\frac{p_n}{p_{n-1}}=1$, therefore, $\frac{p_k-p_{k-1}}{p_{k-1}}\sim\log\left(\frac{p_k}{p_{k-1}}\right)$ Thus, for any $\epsilon>0$, there is an $N$ so that for $m\ge N$,
$$
\begin{align}
m
&=\pi(p_m)\\
&\ge(1-\epsilon)\frac{p_m}{\log(p_m)}\tag{5}
\end{align}
$$
and
$$
\begin{align}
\sum_{k=m+1}^n\frac{p_k-p_{k-1}}{p_{k-1}}
&\le(1+\epsilon)\sum_{k=m+1}^n\log\left(\frac{p_k}{p_{k-1}}\right)\\
&=(1+\epsilon)\log\left(\frac{p_n}{p_m}\right)\tag{6}
\end{align}
$$
Thus, for any $n\ge2N$, there must be a $\frac n2\lt k\le n$ so that
$$
\begin{align}
\frac{p_k-p_{k-1}}{p_{k-1}}
&\le(1+\epsilon)\frac{2\log(2)}{n}\\
&\le(1+\epsilon)\frac{2\log(2)}{k-1}\\
&\le\frac{1+\epsilon}{1-\epsilon}2\log(2)\frac{\log(p_{k-1})}{p_{k-1}}\tag{7}
\end{align}
$$
Comparing $(3)$ and $(7)$, there is no constant of proportionality that would make the given inequality true for all $n$.